The Study Of High-order Finite Difference Method With Satisfying Geometric Conservation Law And Its Applications In Large Eddy Simulations

Although the finite difference method is historically old and the earliest one in de-veloping high-order schemes,there also exist many problems to clarify and study.Based on the high-order scheme WCNS,the present thesis carries out the comparison between the finite difference method and finite volume method,studies the problem of geometric conservation law,conducts the verification of schemes and et al.Finally,representative schemes are applied in the large eddy simulations,which are verified the effectiveness in turbulence simulations as well.The comparison of high-order finite difference and finite volume method is first stud-ied in this thesis.The problem of freestream preservation in the two methods is discussed in detail and the conditions that preserve freestream in finite volume method based on the dimension-by-dimension reconstruction are proposed.By reevaluating the cases in reference,it proves that both of these two algorithms perform well and are comparable to each other in terms of accuracy,resolution of waves and sensitivity to grid irregulari-ties,which corrects the traditional opinion that the finite volume method is better than the finite difference one.Further analysis finds out that the main reason that leads to the in-correct conclusion is the neglect of the problem of geometric conservation law.Therefore a continued study on this problem is carried out and a concept independent of numerical methods and of universal signigicance is proposed.This concept makes a clear clarifica-tion between geometric conservation law and freestream preservation and points out that the geometric conservation law is a basic condition of space that should be satified in the discrete process.Meanwhile some other problems involving in the high-order finte difference method are also studied in this thesis,including flux calculation method,viscous term and et al,which are as well important to numerical schemes.Synthesizing above studies,numerical methods are verified with cases of different dimensions and governing equations,which guarantees the methods and codes.Based on the developed numerical method,high-order schemes are applied in the implicit large eddy simulations.The schemes of HDCS and HWCNS are utilized in this thesis.To verify the effectiveness,two explicit methods of Smagorinsky model and dy-namic Smagorinsky model are also adopted for comparison.Results of typitail cases in large eddy simulations present that the implicit methods based on HDCS and HWCNS schemes have the same ability of simulating turbulence with that of the explicit methods.Furthermore,preliminary analysis on the numerical characteristics of the two methods is made,which provides reference for the further studies.